On a measure theoretic area formula

preprintyear: 2014 journal: Proc. Roy. Soc. Edinburgh Sect. A abstract: We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathe'odory measure. We focus our attention on the case this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our point consists in using certain covering derivatives as ``generalized densities''.
Some consequences for the sub-Riemannian Heisenberg group are also pointed out.